# What do you mean, approaching the speed of light?

If velocity is relative and dependent on an observer then how does an isolated object "approach the speed of light"? Approaching the speed of light relative to what??? Does the ubiquitous constant velocity/closed compartment analogy break down at relativistic speeds? If one were traveling "near the speed of light" and stopped accelerating, what is different about that scenario and any other of constant velocity? I.e. who's to say that 'you' aren't moving at near light speed away from 'me' and not the other way around?
In reality, I think both perspectives may be correct, but if that's so, then why is a near infinite amount of energy required to accelerate further after 'I' have returned to a state of constant velocity 'near the speed of light'? Why doesn't it also take a near infinite amount of energy for 'you' (who we took to be stationary at the start of this exercise) to accelerate in the direction opposite of me?
These questions all really boil down to the same central conundrum. If all constant velocity motion is relative, then how does the concept of velocity have any meaning at all???

## Answers and Replies

phinds
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If velocity is relative and dependent on an observer then how does an isolated object "approach the speed of light"? Approaching the speed of light relative to what??? Does the ubiquitous constant velocity/closed compartment analogy break down at relativistic speeds? If one were traveling "near the speed of light" and stopped accelerating, what is different about that scenario and any other of constant velocity? I.e. who's to say that 'you' aren't moving at near light speed away from 'me' and not the other way around?
In reality, I think both perspectives may be correct, but if that's so, then why is a near infinite amount of energy required to accelerate further after 'I' have returned to a state of constant velocity 'near the speed of light'? Why doesn't it also take a near infinite amount of energy for 'you' (who we took to be stationary at the start of this exercise) to accelerate in the direction opposite of me?
These questions all really boil down to the same central conundrum. If all constant velocity motion is relative, then how does the concept of velocity have any meaning at all???
All motion is relative, but the speed of light is not. You, right now as you read this, are moving at 99.9999999% of c in one frame of reference, 10% of c in another, and not moving at all in your own FoR. None of these FoRs is more "real" than any of the others. Light, on the other hand moves at c in all inertial FoRs.

The energy you are talking about is the energy required to move something that is IN your FoR from stationary (realtive to you), to relativistic speeds (relative to you).

Wow! Thanks for answering so quickly! (I'm new at this... I can accept your explanation and actually understand it to a large degree as I alluded to in my question, but I'm still trying to get my head around its implications. One is that 'potential energy' in the form of inertia becomes relative. What's that do??? There are a ton of other things to think about along these lines so I'm not really expecting a follow up. Thanks again for humoring a newbie... ;)

pervect
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If velocity is relative and dependent on an observer then how does an isolated object "approach the speed of light"? Approaching the speed of light relative to what???
An isolated object doesn't approach the speed of light.
Does the ubiquitous constant velocity/closed compartment analogy break down at relativistic speeds?
No.
If one were traveling "near the speed of light"
relative to some observer which we will call O in some frame of reference that we'll call S
and stopped accelerating, what is different about that scenario and any other of constant velocity? I.e. who's to say that 'you' aren't moving at near light speed away from 'me' and not the other way around?
Nothing. If we call the travelling observer O', O' has an associated frame of reference S' in which O' is stationary, but in which O is moving.

In reality, I think both perspectives may be correct, but if that's so, then why is a near infinite amount of energy required to accelerate further after 'I' have returned to a state of constant velocity 'near the speed of light'? Why doesn't it also take a near infinite amount of energy for 'you' (who we took to be stationary at the start of this exercise) to accelerate in the direction opposite of me?
These questions all really boil down to the same central conundrum. If all constant velocity motion is relative, then how does the concept of velocity have any meaning at all???

The explanation that "it takes a nearly infinite amount of energy" is in fact true, because energy is relative as well as velocity. However, as you've noticed, it doesn't actually give a lot of insight as to why one can't reach the speed of light, or what happens when one tries.

What is more relevant is to look at how velocities add in special relativity. In the above scenario, we have an observer O, and an observer O' moving at some velocity v1 relative to O. Let us introduce a new observer, O'', moving at some velocity v2 relative to O', and ask what the velocity of O'' is relative to O.

Example: Suppose v1 is 1 meter/second less than the speed of light, i.e. 299 792 457 meters/second. And suppose v2 is 10 meters per second. Then O'' is moving at 10 meters per second relative to O', and O' is moving at 299 792 457 meters/second relative to O.

Now we ask - how fast is O'' moving relative to O? The answer is that O'' is still moving less than the speed of light, because velocities don't add. The formula for velocity addition is

##v_{tot} = \frac{v1+v2}{1 + v1\,v2 / c^2}##

Working out the example, we get in the above example v_tot ##\approx## 299 792 457.000 000 066

So we see an alternate (and I think better) explanation of why one can't reach the speed of light. It's due to the way velocities add in special relativity. This is a more satisfying answer I think, , one reason it is more satisfying is that it does not involve the extraneous issues of dynamics in a question that can be answered purely with kinematics.

Great response. I appreciate the detail. Kinematics and dynamics in the physical sense weren't a strong part of my curriculum in ChemE, but I think I do understand your answer. In the vector calculus problem regarding O, O', and O", traditional mathematics would guarantee that the velocity of O" must be greater than the speed of light because velocities *are* additive. But that neglects the second term in the denominator, which under "normal" conditions (excuse the self-centeredness, but you know what I mean :) ) is a perfectly reasonable assumption, but at relativistic speeds becomes significant. Wow! That follows pretty obviously from the equation you gave. Does the theory provide any insight as to why this equation would describe the universe so well?

Ooohh I think I remember what kinematics and dynamics are in math. Algebra vs. Calculus, right?

And what about adding to your example above an O'" traveling at velocity v3 in relation to O"? How does the equation change?

Orodruin
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One is that 'potential energy' in the form of inertia becomes relative.
Energy related to movement is kinetic enery, not potential. Inertia at relativistic speeds is also dependent on the direction of acceleration and you cannot ascribe an energy to it.

In general, energy is a frame dependent quantity and you should not expect it to be the same in different frames. This is true also in Newtonian physics.

pervect
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Kinematics is the branch of classical mechanics which describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without consideration of the causes of motion.

Thus kinematics does not include forces - it does describe velocities, accelerations, and so forth, as those describe how objects move.

One of the simpler ways to describe how velocities add in SR is the concept of rapidity. The wiki article is at http://en.wikipedia.org/w/index.php?title=Rapidity&oldid=641472354

In one (spatial) dimension, the relative rapidity between two objects is the hyperbolic arc tangent , ##\tanh^{-1}##, of v/c, v being the relative velocity. Rapidities (in 1 dimension) add - so if you have a chain of n objects all moving in the same direction, the rapidity between the first and last object in the chain is the sum of the rapidities between the different objects in the chain.

The wiki article has a short (though somewhat advanced) proof of how to derive the addition of rapidities from the Lorentz transform. The Lorentz transform describes the the relationship between the coordinates of an object as seen from different frames of reference. There is a 1:1 mapping between the coordinates (t,x) which describe the position of an object in some frame S, and (t', x') which describe the position of the object in some other frame S'. This mapping is called the Lorentz transform, I wont give the formulas here, but a bit of research should turn them up. For a simple derivation of the Lorentz transform from the basic principles of relativity, I'd recommend Bondi's rather old book, "Relativity and Common Sense", aviailable on the internet archive at https://archive.org/details/RelativityCommonSense

Fredrik
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If velocity is relative and dependent on an observer then how does an isolated object "approach the speed of light"? Approaching the speed of light relative to what???
Relative to the observer. (What this really means is "according to the coordinate assignments made by the coordinate system we associate with the observer's motion). The speed of the object relative to inertial observer A will typically be different from the speed of the object relative to inertial observer B. The speed of light relative to inertial observer A will however always be the same as the speed of light relative to inertial observer B, exactly c=299792458 m/s.

Does the ubiquitous constant velocity/closed compartment analogy break down at relativistic speeds?
If you're referring to the idea that experiments in a (non-accelerating) room sealed off from the environment will have the same result regardless of the velocity of that room (relative to some observer), then no, it doesn't break down. It's equally valid at all speeds less than c.

If one were traveling "near the speed of light" and stopped accelerating, what is different about that scenario and any other of constant velocity? I.e. who's to say that 'you' aren't moving at near light speed away from 'me' and not the other way around?
Nothing and no one. If your velocity in my coordinate system is 0.99c, then my velocity in yours is -0.99c.

In reality, I think both perspectives may be correct, but if that's so, then why is a near infinite amount of energy required to accelerate further after 'I' have returned to a state of constant velocity 'near the speed of light'? Why doesn't it also take a near infinite amount of energy for 'you' (who we took to be stationary at the start of this exercise) to accelerate in the direction opposite of me?
If you switch on your rocket engine to accelerate away from me, then our descriptions of what's going on are no longer similar. You will feel like you're getting pushed toward the floor. I won't feel anything like that. "My coordinate system" is still an inertial coordinate system, but it's impossible to associate an inertial coordinate system with your motion. We can associate one with each event you experience, but it would be a different one at each event.

These questions all really boil down to the same central conundrum. If all constant velocity motion is relative, then how does the concept of velocity have any meaning at all???
Look at a spacetime diagram, and in particular at the straight lines through the origin. Straight lines represent constant-velocity motion. A Lorentz transformation map the lines x+ct=0 and x-ct=0 to themselves, but all other straight lines through the origin are mapped to straight lines through the origin with a different slope (=different velocity).

Thanks for the reply. I'm obviously a bit new at this and probably need a basic physics refresher. I didn't quite follow the explanation of the spacetime diagram, but I do think I get the gist of what they represent even if I can't articulate it very well.

Energy related to movement is kinetic enery, not potential. Inertia at relativistic speeds is also dependent on the direction of acceleration and you cannot ascribe an energy to it.

In general, energy is a frame dependent quantity and you should not expect it to be the same in different frames. This is true also in Newtonian physics.
Well I feel a bit dumb with the potential vs kinetic mixup. It's been awhile since my underclassman physics classes... Lol
Now that you mention it though I do remember that establishing a frame of reference is one of the first things you have to do in a lot of physics problems, static, kinematic, or dynamic I guess. Thanks for the correction!

I'd recommend Bondi's rather old book, "Relativity and Common Sense", aviailable on the internet archive at https://archive.org/details/RelativityCommonSense
Thanks for the response in total and especially the book recommendation. I have a lot of reading to do if I'm ever really going to understand this stuff. :)

Fredrik
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Thanks for the reply. I'm obviously a bit new at this and probably need a basic physics refresher. I didn't quite follow the explanation of the spacetime diagram, but I do think I get the gist of what they represent even if I can't articulate it very well.
I found it difficult to answer that question, mainly because I don't see why one would conclude that velocity is meaningless.

The first two chapters of "A first course in general relativity" by Schutz is an excellent introduction to special relativity. His presentation is built up around spacetime diagrams.